By themselves, these equations are not a complete theory of electromagnetism. In particular, they lack any information on how material particles, such as electric charges and magnets, respond to the fields E and B. To use a Judeo-Christian analogy, Maxwell’s equations represent the state of the universe after God said “Let there be light,” and before he created anything else. To incorporate the material world, Maxwell added extra terms (representing charge density and current density) and extra equations.
Most of Maxwell’s equations were not actually original to him. The individual equations are known as Gauss’s law, Faraday’s law, and Ampère’s law. Maxwell’s only new contribution was a correction term that enters Ampère’s law when electric currents are taken into account. Nevertheless, the understanding that the equations could be brought together into a system, and the idea that magnetic and electric fields were the fundamental agent, were entirely due to Maxwell. So, too, was the discovery that the speed of light, c—the only experimental constant to be found in these equations—is a fundamental physical law.
Earlier it was mentioned that Euler’s equation eiπ + 1 = 0 was voted the most beautiful equation of all time by readers of Mathematical Intelligencer. In 2004, Physics World conducted a similar poll. It is no surprise that the readers of that publication chose Maxwell’s equations as the greatest ever. They are so simple, so symmetric, so hard-earned, and they explain so much.
Yet like the other revolutions described in this chapter, they made little impression at the time. Maxwell’s contemporaries just didn’t know what to make of them. “As long as I cannot make a mechanical model all the way through I cannot understand; and that is why I cannot get the electromagnetic theory,” said William Thomson, Lord Kelvin, in 1884 (the same Lord Kelvin who couldn’t “get” quaternions!).
But over time, the significance of Maxwell’s equations became more apparent. They predicted that electromagnetic waves could exist with different wavelengths—such as the waves we now call microwaves, infrared, ultraviolet, and X-rays. They predicted that such waves could be created by oscillating electric fields. In 1901, Guglielmo Marconi used precisely this principle to transmit the first radio waves. They implied that light itself can exert pressure. Sure enough, researchers in the twentieth century discovered the “solar wind,” which explained the centuries-old mystery of why comet tails point away from the Sun. And in 1905, as will be discussed further in the next chapter, they led Albert Einstein to the theory of relativity.
* * *
‡ A transverse wave is one that propagates at right angles to the motion of the individual particles in the wave. An example is “the wave” (sometimes called “the Mexican wave”) at a sports stadium. The individual particles (i.e., the fans) go up and down, but the wave moves around the stadium.
PART FOUR
equations in our own time
No scientist is more emblematic of the twentieth century than Albert Einstein. One part imp and one part prophet, he sticks out his tongue at us in one famous photograph, and looks at us with world-weary eyes in another. His unruly hair and lack of concern for social conventions have come to define the popular image of science. He was the world’s first scientist-as-rock-star.
In some ways science could not have asked for a better ambassador to the public. Einstein’s fame was truly deserved. Not just once, but over and over again, he transformed the worldview of physicists. He was the first physicist to understand the quantization of light; he was the first to recognize the equivalence of matter and energy; his name is synonymous with the theory of relativity. Yet he also transcended science. He used his fame to advance the cause of pacifism, at least until the rise of Nazi Germany made that position untenable for him. In 1940, he warned President Franklin Roosevelt about the threat posed by a possible atomic bomb, a warning that paved the way for the Manhattan Project and profoundly affected the balance of power in the postwar world.
Einstein’s discoveries were just as much a result of his personality and the time in which he lived as of his intellect. By nature he loved to question authority. While other physicists were hesitant to discard centuries of tradition, Einstein was completely unconcerned and even happy to do so. He was lucky to come of age at a time when physicists had three mature, thoroughly understood theories—mechanics, electromagnetism, and thermodynamics—that fundamentally contradicted each other in subtle ways. While others averted their eyes from the contradictions, Einstein dared to look straight at them and pointed out how to overcome them.
Curiously, Einstein was not a lover of mathematics early in his career. His former math teacher Hermann Minkowski once wrote, “In his student days Einstein had been a lazy dog. He never bothered about mathematics at all.” But Einstein’s attitude completely changed over time. Minkowski’s mathematical reformulation of special relativity, in 1908, helped his theory win acceptance. Einstein could never even have written down his theory of general relativity without a deep understanding of non-Euclidean geometry. By 1912, Einstein had recanted his former disdain: “I have gained enormous respect for mathematics, whose more subtle parts I considered until now, in my ignorance, to be pure luxury!”
Sometimes converts make the best missionaries. Even if Einstein was a reluctant mathematician, he certainly enhanced the prestige of the subject. Thus it makes sense to start our history of the equations of the twentieth century with him.
19
the photoelectric effect quanta and relativity
The greatest revolution in twentieth-century physics began with a seemingly insignificant observation. In 1887, the German physicist Heinrich Hertz noticed that he could get a spark to jump between two metal electrodes more easily when the electrodes were exposed to light (specifically ultraviolet light) than he could if they were in the dark.
Another German physicist, Philipp Lenard, showed in 1902 that shining a light on a metal caused the metal to emit what were then known as “cathode rays,” and are now known as electrons. If the electrons had sufficient energy, they could produce the sparks that Hertz had seen. The phenomenon became known as the photoelectric effect: the production of electricity from light. By varying the intensity and frequency (i.e., the color) of the light, Lenard discovered some strange things about the photoelectric effect. It does not occur at all with red light, no matter how intense the light is. Also, the energy of the emitted electrons does not increase when the intensity of the light increases—only when the frequency increases.
In 1905, Albert Einstein, who was at that time a 26-year-old patent clerk in Bern, Switzerland, proposed a revolutionary explanation for Lenard’s discovery. He hypothesized that light “behaves like a discontinuous medium consisting of energy quanta.” His “energy quanta” are now called photons. Einstein argued that each photon contains a characteristic amount of energy, which is proportional to the frequency, ν, of the light: E = hν.
E represents the energy of an object; m its mass at rest; and c = 299,792,458 meters per second is the speed of light. Einstein’s equation implies that matter is a form of energy.
The quantization of light means that the absorption of a photon is an all-or-nothing deal. When a photon hits the surface of the metal, it cannot be half absorbed and half reflected (as an ordinary wave can). If it is absorbed, all of its energy goes into the target. If that energy exceeds the binding energy (P) that holds electrons to the metal surface, then the metal will emit an electron with a characteristic energy hν – P.
Einstein’s light-quanta hypothesis explains why red light fails to induce a photoelectric effect. Red light has a longer wavelength, and a lower frequency, then green or blue light. Because their frequency ν is smaller, the photons in red light do not have enough energy to kick out an electron. (In other words, hν < P.) If you make the light more intense while keeping the color the same, the number of photons will increase, but none of them are energetic enough to produce the photoelectric effect.
The strange behavior of photons has real consequences
. For example, in recent years, there has been controversy over an alleged link between mobile phone usage and cancer. Holding a cell phone next to your head increases the intensity of the radiation that you are exposed to. Therefore, it seems like common sense that this will increase the risk of radiation-induced damage to your cells. However, this is completely wrong. If the general public understood Einstein’s century-old discovery of the quantization of light, the mobile-phone health scare would never have gotten started.
Radiation damages biological tissues by knocking electrons loose from atoms, making the atoms more reactive. Just as in the photoelectric effect, it is the frequency of the radiation that matters, not the intensity. The infrared radiation emitted by a cell phone has even lower frequency than red light, so it is below the threshold where it can knock an electron loose from a metal atom. Moreover, our bodies are not made of metal! The atoms in our bodies hold their electrons more tightly than metals do. For both of these reasons, infrared light is completely safe to us; it cannot ionize the atoms in our body. Red, green, and blue light are also perfectly safe. Otherwise we would have to live in caves, and avoid exposing ourselves to green grass and a blue sky.
Opposite Atomic burst over Hiroshima, from the first atomic bomb used in military action, nicknamed “Little Boy.”
We need to start worrying about cancer only when we are exposed to higher frequencies of radiation, such as ultraviolet light or X-rays. If the frequency is high enough to ionize the atoms in your body, then the intensity will start to matter—but not until then.
FOR THE PHYSICISTS of Einstein’s day, the light-quanta hypothesis flew in the face not only of common sense, but also of a century of theory. The debate over whether light was a particle or a wave had been going on since the early 1800s, and had apparently been resolved in favor of waves. Maxwell’s equations proved that light is an electromagnetic wave.
Now Einstein was reopening a question that had seemed to be settled for good. Older physicists did not take kindly to it. In 1913, Max Planck wrote, “[Einstein] might sometimes have overshot the target in his speculations, as for example in his light quantum hypothesis.” In 1916, Robert Millikan wrote that the hypothesis “may well be called reckless.” Nevertheless, Millikan’s own experiments showed that Einstein was correct, and six years later Einstein won the Nobel Prize for his explanation of the photoelectric effect.
In fact, Einstein’s equation E = hν explained much more than one minor experimental effect. It was the first shot in the quantum revolution. Quantum mechanics resolved the age-old “particle versus wave” debate in an utterly paradoxical fashion. Light is both a particle and a wave. Which one it “looks like” depends on how you interrogate it. If you measure its frequency and wavelength, light looks like a wave. If you count photons by using the photoelectric effect, then light looks like a particle.
Any attempt to describe this wave-particle duality in common language tends to fall flat, because there is nothing like it in our experience of the macroscopic world. Intuition and common sense often mislead us, as the cell-phone example shows. In the subatomic world, mathematics is the only reliable guide.
If Einstein had been any ordinary scientist, or even any ordinary Nobel laureate, his equation E = hv would have been the achievement of a lifetime. Instead, it is not even his most famous equation beginning with the letter E! That distinction, of course, falls to another equation that he first set down on paper in 1905:
This equation expresses, with elegant simplicity, the equivalence between matter and energy. A particle of mass m contains an amount of energy E that is equal to the product of the mass and the square of the speed of light (c2). Because the speed of light is a gigantic number, even a tiny amount of matter can be converted to vast amounts of energy. The bombing of Hiroshima in 1945 proved this only too well. The “Little Boy” bomb contained about 140 lb (64 kg) of enriched uranium. The amount of matter actually transformed into energy in the explosion was a little more than the weight of one BB pellet. One BB pellet was enough to destroy an entire modern city.
When Einstein discovered the equivalence of energy and mass, in 1905, he did not yet foresee its terrible consequences. He wrote to a friend: “The principle of relativity in relation to Maxwell’s equations demands that mass is a direct measurement of the energy of a body; that light carries mass. A noticeable decrease in mass must then occur in the case of radium. The thought is funny and infectious; but whether God is laughing and has led me by the nose, I do not know.” Forty years later, the whole world found out that God was not laughing.
How did Einstein arrive at his most famous equation? It goes back to his second great discovery of 1905: the theory of special relativity.
NEWTON’S LAWS OF MOTION hold true in any inertial frame of reference—that is, one moving at a constant velocity. If you are traveling in a train at a constant speed of 100 meters per second, it feels just the same to you as if the train were standing still and the surrounding landscape were rushing by you at 100 meters per second. No physical experiment can tell you the difference. On the train, bodies moving in a straight line will continue to move in a straight line (Newton’s first law). On the train, an applied force will produce an acceleration according to the equation F = ma (Newton’s second law). On the other hand, if the train suddenly slows down or speeds up, so that its velocity is not constant, you can detect this fact.
However, Maxwell’s equations for electromagnetism do not seem to behave the same way. Recall that the speed of light appears in Maxwell’s equations as a physical constant. Therefore, if the relativity principle applies to Maxwell’s equations, any measurement of the speed of light in any inertial reference frame should give the same result—299,792,458 meters per second.
But that fact leads to a paradox. If a car is traveling at 120 meters per second, and you follow it on a train traveling at 100 meters per second, then it should seem to you as if the car is going much slower—only 20 meters per second. Likewise, if you are chasing a light wave, it should appear to move 100 meters per second slower than its normal speed (that is, 299,792,358 meters per second). Or if you are approaching a light wave head on, it should appear to be moving 100 meters per second faster than usual.
In fact, we do, sort of, live on a moving train—we call it Earth. Because our orbit around the Sun takes us in different directions at different times of year, physicists reasoned that the velocity of light coming from a distant star should appear to change, depending on whether we are moving toward or away from it. Yet many experiments, including a famous one by Albert Michelson and Edward Morley in 1887, failed to discover any such changes. Einstein learned about these experiments while he was a student.
Einstein argued that the experiments had failed because there is nothing to detect. He elevated the relativity principle to the status of a postulate: the laws of physics are the same in all inertial reference frames. In particular, this means that speed of light is a universal constant. We do not have to abandon Maxwell’s equations; nor do we have to abandon Newton’s laws. We do, however, have to modify them. The common-sense subtraction of velocities, given in the example above, is incorrect and has to be replaced by a somewhat more elaborate formula. More importantly, we have to abandon our common-sense conceptions of space and time. According to Einstein, the reason we do not detect any change in the speed of light is that lengths and time intervals are relative. They depend on your frame of reference.
Below Albert Einstein, (1879–1955).
Imagine you are on a train speeding past a stationary Albert at 100 meters per second. As your train goes whizzing by, Albert (if he is very observant) will notice that it has gotten a little bit shorter than it was when it was standing still, and he will also notice that the watch on your wrist is running a little bit slower. If you synchronize your watches so that they both read 10:00 the instant that you pass him in your train car, he will see his watch reach 10:01 before yours, because of the time dilation effect. But you will insist
that your watch reached 10:01 first! And you are both right. By the time that the light waves from his digital watch reach you, showing the readout 10:01, your watch will already show 10:01, and vice versa. In Einstein’s universe, there is no absolute measurement of time, and in fact there is no absolute concept of “before” and “after.”
The only reason we do not normally perceive the shrinking of space or the dilation of time is that we normally move at very slow speeds compared with light. Thus the effects are extremely small. However, with precision instrumentation it is possible to test the predictions of relativity theory. A clock launched into orbit and then brought back to Earth really does run a few nanoseconds slow. Global positioning satellites take relativistic effects into account. Your GPS receiver compares time signals received from several different satellites, in order to determine how far away you are from them. Each of those satellites is moving rapidly with respect to you, so their clocks will be slowed down by the time dilation effect. Thanks to GPS, we are now living in the era of applied relativity.
EINSTEIN FORMULATED two different theories of relativity, as mentioned earlier. In “special relativity,” which he developed in 1905, he assumed that the laws of physics are the same when viewed from any inertial reference frame (i.e., one moving at constant velocity). However, it continued to bother Einstein that accelerated reference frames (which move at non-constant velocities) were somehow different. For several years, he sought a truly general theory of relativity in which the laws of physics would be expressed in the same way regardless of the observer’s frame of reference.
The Universe in Zero Words Page 13
